First-passage method for the study of the efficiency of a two-channel reaction on a lattice

We study the efficiency of a two-channel reaction between two walkers on a finite one-dimensional periodic lattice. The walkers perform a combination of synchronous and asynchronous jumps on the lattice and react instantaneously when they meet at the same site (first channel) or upon position exchange (second channel). We develop a method based on a conditional first-passage problem to obtain exact results for the mean number of time steps needed for the reaction to take place as well as for higher order moments. Previous results obtained in the framework of a difference equation approach are fully confirmed, including the existence of a parity effect. For even lattices the maximum efficiency corresponds to a mixture of synchronous events and a small amount of asynchronous events, while for odd lattices the reaction time is minimized by a purely synchronous process. We provide an intuitive explanation for this behavior. In addition, we give explicit expressions for the variance of the reaction time. The latter displays a similar even-odd behavior, suggesting that the parity effect extends to higher order moments.